| Blog Entry: Einstein’s field |
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| Written by Sean Miller | |
| Monday, 23 October 2006 | |
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The following is an essay that explores Einstein's theory of general relativity as a conceptual metaphor: With the theory of general relativity, Einstein articulated the fundamental equivalency between acceleration[1] and the force of gravity within the spacetime framework of special relativity. A famous equation, now called the Einstein Field Equation, expresses this relationship:
G stands for the gravitational field tensor. T represents the stress-energy tensor. π (pi) is 180 degrees of rotation along the unit circle. a and b codify an arbitrarily determined movement from point a to point b along the four-dimensional spacetime metric. Metaphorically speaking, a tensor is a container that holds a collection of attributes that, in the formal mechanism, are associated with particular coordinates within the geometric space of the system. The values of these tensor attributes are functions of motion, both momentum and acceleration, through this space from one set of coordinates to another. In this sense, the tensor algebra of general relativity transforms geometric spaces into informational spaces. The topology of the geometry reflects a smoothly transforming collection of information that encodes densities of energy relating to momentum and acceleration. Unlike special relativity—with its flat Minkowski space, the geometry of general relativity is Riemannian: this informational space, with its basket of information having to do with stress-energy, with the density and flux of energy within the system, possesses curvature—it bends and warps in dynamic response to the motions and densities of energy and matter that constitute it.[2]
The basic-level metaphor most pertinent to general relativity would then be the field, metonymically represented by G in the above formula. The essence of the general relativistic universe is energy: energy manifests as the spacetime field and its patterns of change are curvature of the spacetime field itself. As such, change is motion where acceleration equates to the force of gravity. More specifically, change is turning, where in the source-path-goal schema of general relativity, a body’s motion in spacetime follows a curved trajectory. As with special relativity, time is a space-like dimension. The force of gravity, then, is curvature of space—a considerable revision and depersonalization of Newton’s theory, which, as discussed above, describes gravity metaphorically as the invisible hands of rigid bodies pulling each other together. In general relativity, the earth does not pull the moon toward it such that, accounting for the motion and momentum of the moon’s trajectory, it assumes an elliptical orbit around the earth. In general relativity, due to the earth’s gravity—the energy-density of its mass, a divot forms around it in spacetime. The moon falls along the curvature of this divot towards the earth, like the ball whipping around a roulette wheel. The force of gravity then, is a falling towards, rather than a pulling in. Rigid bodies with invisible hands give way in general relativity, to a divot-like energy density sending out ripples of force. As such, unlike Newtonian gravity, which is instantaneous, general relativity shows that gravity, like all forms of energy, must obey the speed limit set by light. Yet general relativity accommodates the Newtonian formalism. Under conditions consistent with the descriptive reach of Newton’s theory, the Einstein Field Equation reproduces its results: it is backwards compatible. All the experimental evidence that confirms Newtonian mechanics also serves general relativity. Of course, since its scope is much broader, general relativity makes more precise predictions, in addition to making predictions beyond the reach of Newtonian mechanics (like the expanding universe). General relativity is well suited to massive scales: to the interactions of massive bodies across vast swaths of spacetime—stars, galaxies, the entire universe—where the microscopically negligible force of gravity becomes macroscopically dominant. As such, the causality of general relativity has a location-event structure where the ground, curved spacetime, takes precedence over the figure, massive bodies such as stars and black holes. Causation, then, is the forced movement of an entity to a new location within the tensor geometry. Massive bodies fall towards each other: this is seen in the theory as curves and warps in the spacetime whole, pockets of dense energy. States are locations; and changes of state are movements from one location to another within the informational space (recall that locations in this space carry a basket of attributes, the energy-density tensor). Natural causes, therefore, are motions to and from locations. The closest to a human agent the theory of general relativity upholds would be the energy that constitutes the gravitational field. Energy is the thread that, woven together, forms the fabric of the spacetime field. In special relativity, bodies expand, grow plastic, and open their eyes—they sprout unique points of view; with general relativity, it is the energy of the universe that, in a certain metaphoric sense, both watches and constitutes itself. One of the leading contemporary experts on general relativity, Lee Smolin, argues for an even more emphatic apotheosis of the gravitational spacetime field. In his interpretation, general relativity teaches us that the essence of the universe is a particular pattern of change more fundamental than motion along spacetime curvature: The metaphor in which space and time together have a geometry, called the spacetime geometry, is not actually very helpful in understanding the physical meaning of general relativity. […] Our world cannot be understood as a collection of independent entities living in a fixed static background of space and time. Instead, it is a network of relationships the properties of which are determined by its relationships to the other parts.[3]
The rigid bodies of the Newtonian universe, where matter holds the attention of the one absolute observer, dissolve into energy-densities: curves, warps, and divots within the spacetime whole. The duality of figure and ground give way to only ground: the theory realizes the universe not as composed of myriad bodies (‘entities’) hovering above a ‘fixed static background’, but rather, as transitory, contingent forms emerging from the substance of spacetime, its energy flux. As such, bodies gain their existence solely through patterns of change, through the ‘network of relationships’ that compose the totality. This conception of the essence of the universe as a network of relationships finds its complement, on a microscopic scale, in quantum theory, the subject of the next case study. [1] Note: Velocity measures the motion of a body along an axis in space: metres per second. Momentum, as a force, represents the product of rest mass and velocity. Acceleration measures the rate of increase in velocity with respect to time, as in, for example, a massive body falling to earth, measured as a constant acceleration of approximately 9.8 metres per second squared. [2] The ‘8π’ in the equation provides a clue to this feature: four groups of 360 degree (2 x 180 degrees) rotations for the four unit circles of the four spacetime dimensions. Riemannian geometry exploits i, the imaginary number of the square root of negative one, a mathematical concept that describes rotation about a unit circle, allowing, as a consequence, curvature. [3] Lee Smolin, Three Roads to Quantum Gravity (New York: Basic Books, 2001), pp. 59, 63. |
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Furthermore, unlike the uniformly fixed metric of 
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